This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340566 #19 Jan 12 2021 02:43:00
%S A340566 0,11,1,111,1110,11101,1110100,111010,10,1100,100,1110101,110,1111,
%T A340566 11100,1110111,101,11001,111011,11111,1101,110011,1010,11101001,1000,
%U A340566 11101011,111010010,10001,1110000,111110,1110110,111000,11000,111010110,11110,111010000
%N A340566 Square array, read by descending antidiagonals; T(n,k) is A001057(n) + A001057(k)*i, converted to complex binary (base -1 + i), where i=sqrt(-1).
%C A340566 Complex binary (base -1 + i) has the ability to express all positive or negative, real or complex, integers with only 2 numerical symbols ('0' and '1') as integers, without the need for a sign marking the integers as such.
%C A340566 Converting a real number, n, to complex binary requires one to convert it to base -4 ((n + N) xor N, N = floor(4/5*16^(ceiling(log_4(abs(n))) + 1))), then adding 10 to every digit greater than 1, then treating it as a number in base 16 and converting that to binary. (E.g., -5 => [2,3] => [12,13] => 205 => 11001101.)
%C A340566 Converting a complex number, n + k*i, requires one to convert X = n + k into complex binary and then convert k into the same but shift it one digit to the left. After this, one must add them together. This functions much the same way as binary addition, but the carry is '110' rather than '1' and 11 + 111 = 0.
%D A340566 T. Jamil, Complex Binary Number System, Springer, 2013.
%e A340566 Square array T(n,k) begins:
%e A340566 \k 0 1 2 3 4 5 6 ...
%e A340566 n\
%e A340566 0| 0 11 111 1110100 100 1110111 110011 ...
%e A340566 1| 1 1110 111010 1110101 101 1010 111110 ...
%e A340566 2| 11101 10 110 11001 11101001 1110110 110010 ...
%e A340566 3| 1100 1111 111011 1000 111000 1011 111111 ...
%e A340566 4| 11100 11111 11101011 11000 11101000 11011 11101111 ...
%e A340566 5| 1101 111010010 111010110 1001 111001 1100110 100010 ...
%e A340566 6| 10001 11110 11101010 1110100101 10101 11010 11101110 ...
%o A340566 (PARI) A340566(n,k)={my(A001057(x)=if(x%2,x\2+1,-x/2),V=vecsum(Vec(matconcat(apply(w->my(Y=if(w,A001057(k), A001057(n)+A001057(k)));if(Y, my(X=floor(4^(2*logint(abs(Y), 4)+5)/5));Vecrev(binary(shift(fromdigits(apply(z->z+(10*(z>1)), digits(bitxor(Y+X,X),4)),16),w)))),[0,1])~)~))~);
%o A340566 while(vecmax(V)>1,my(Z=Vec(select(x->x>1,V,1)));for(x=1,#Z,my(z=Z[x]);if(V[z]<=1,,(z+2<=#V)&&(V[z+1]>1)&&V[z+2],for(j=z,z+2,V[j]-=2^(j!=(z+2))),(z+4<=#V)&&vecmin(V[z+2..z+4]),V[z]-=2;for(j=z+2,z+4,V[j]-=1),z+1>#V,V[z]-=2;V=concat(V,[0,1,1]),V[z]-=2;for(j=z+2,z+3,if(j<=#V,V[j]+=1,V=concat(V,1))))));fromdigits(Vecrev(V))}
%o A340566 (PARI) { T(n,k) = my(z=n\/2*-(-1)^n + k\/2*-(-1)^k*I, ret=List([]));
%o A340566 while(z, my(bit=(real(z)+imag(z))%2);
%o A340566 listput(ret,bit); z=(z-bit)/(I-1));
%o A340566 fromdigits(Vecrev(ret)); } \\ _Kevin Ryde_, Jan 12 2021
%Y A340566 Cf. A001057, A318438, A318439.
%K A340566 nonn,tabl,base
%O A340566 0,2
%A A340566 _Davis Smith_, Jan 11 2021